Hi Matt, welcome to Outdoorking.
The first thing we need is the information stamped on the cooling air cowl of that Briggs engine. It will be in three groups, labelled Model, Type, and Code, and will probably be on the vertical surface near the air cleaner. That will give us the engine's date of manufacture, and if it is the original engine, the date of manufacture of the mower will probably be just a few months later. In most cases, that same information will give us access to the engine's Illustrated Parts List, to find the part numbers for any replacement parts.
If the smoke it is blowing is blue, chances are you need to replace the piston rings. It would be unusual to need to replace the piston unless the engine has run a whole lot of hours and the rings have already been replaced at least once, or it has been seized up at some time (either by running out of oil, or having its cooling fins blocked up with grass, wasp nest, etc.)